### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

p(0) → 0
p(s(x)) → x
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0, s(y)) → 0
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0), s(s(y))) → 0
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
minus(s(x), s(y)) →+ minus(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), y / s(y)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

p(0') → 0'
p(s(x)) → x
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0', s(y)) → 0'
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) → 0'
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(x)) → x
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0', s(y)) → 0'
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) → 0'
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
log :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
minus, div, log

They will be analysed ascendingly in the following order:
minus < div
minus < log
div < log

### (8) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(x)) → x
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0', s(y)) → 0'
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) → 0'
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
log :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
minus, div, log

They will be analysed ascendingly in the following order:
minus < div
minus < log
div < log

### (9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n40)

Induction Base:
minus(gen_0':s2_0(0), gen_0':s2_0(0)) →RΩ(1)
gen_0':s2_0(0)

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) →RΩ(1)
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) →IH
gen_0':s2_0(0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (11) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(x)) → x
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0', s(y)) → 0'
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) → 0'
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
log :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
div, log

They will be analysed ascendingly in the following order:
div < log

### (12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol div.

### (13) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(x)) → x
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0', s(y)) → 0'
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) → 0'
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
log :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
log

### (14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol log.

### (15) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(x)) → x
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0', s(y)) → 0'
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) → 0'
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
log :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n40)

### (18) Obligation:

TRS:
Rules:
p(0') → 0'
p(s(x)) → x
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
minus(x, s(y)) → p(minus(x, y))
div(0', s(y)) → 0'
div(s(x), s(y)) → s(div(minus(s(x), s(y)), s(y)))
log(s(0'), s(s(y))) → 0'
log(s(s(x)), s(s(y))) → s(log(div(minus(x, y), s(s(y))), s(s(y))))

Types:
p :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s
log :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n40)